### §10. Parabolic Fixed Points: The Leau-Fatou Flower

Again we consider functions *f*(*z*) = λ*z* + *a*_{2}*z*^{2} + *a*_{3}*z*^{3} +... which are defined and holomorphic in some neighborhood of the origin, but in this section we suppose that the multiplier λ at the fixed point is a root of unity, λ^{q} = 1. Such a fixed point is said to be *parabolic,* provided that *f*^{q} is not the identity map. (Compare Lemma 4.7.) First consider the special case λ = 1. Then we can write our map as

with *n* ≥ 1 and *a* ≠ 0. The integer *n* + 1 is called the *multiplicity* of the fixed point. (Compare Lemma ...